The inverse hyperbolic cotangent calculator computes arcoth(x) (also written as cothโปยน(x)) for any value with |x| > 1. The formula is arcoth(x) = ยฝ ln((x+1)/(xโ1)). This is the inverse of the hyperbolic cotangent function coth(t) = cosh(t)/sinh(t).
Inverse Hyperbolic Cotangent Calculator
Domain: |x| > 1 (x < โ1 or x > 1)
Result
Calculatedarcoth(2)
0.5493
Result0.5493
Logarithmic Formยฝ ln(3)
Formulaยฝ ln((x+1)/(xโ1))
What is Inverse Hyperbolic Cotangent?
Inverse hyperbolic cotangent (arcoth or cothโปยน) returns the value whose hyperbolic cotangent equals the input.
- Domain: |x| > 1 (x < โ1 or x > 1)
- Range: all real numbers except 0
- Formula: arcoth(x) = ยฝ ln((x+1)/(xโ1))
- Relationship: arcoth(x) = artanh(1/x) for |x| > 1
- Derivative: d/dx[arcoth(x)] = 1/(1 โ xยฒ) for |x| > 1
How to Calculate Inverse Hyperbolic Cotangent
Calculating the inverse hyperbolic cotangent (arcoth) evaluates specific functions spanning precisely outside the normal domain boundaries associated with traditional tangency.
Computation software strictly resolves arcoth via a slightly modified natural logarithmic structure:
arcoth(x) = (1/2) * ln((x + 1) / (x − 1))
Similar to standard mathematical symmetries, the derivative equation mimics its counterpart identically structurally: d/dx [arcoth(x)] = 1 / (1 − x²), but restricted purely to domains surpassing absolute 1.
Difference between Hyperbolic Cotangent and its Inverse
- Hyperbolic Cotangent (coth): Defines geometries mapping to exponential decay tracking outward exclusively reaching but never interacting directly with positive/negative boundaries.
- Inverse Hyperbolic Cotangent (arcoth): Translates external domain fractions (greater than 1 or less than −1) accurately back alongside a coordinate phase shift parameter structure.
Real-Life Applications
Arcoth maintains significant traction within unique edge cases of applied science fields:
- Magnetic Resonance: Used accurately when detailing absolute ferromagnetic relaxation times in specific physical fields spanning external field interactions.
- Electromagnetic Wave Theory: Modeling voltage phase shift frequencies traveling down heavily distorted infinite medium transmission lines.
Common Values
Click any row to calculate.
| x | arcoth(x) | Exact Form |
|---|---|---|
| โ5 | โ0.2027 | ยฝ ln(2/3) |
| โ2 | โ0.5493 | โยฝ ln(3) |
| 1.5 | 0.8047 | ยฝ ln(5) |
| 2 | 0.5493 | ยฝ ln(3) |
| 3 | 0.3466 | ยฝ ln(2) |
| 5 | 0.2027 | ยฝ ln(3/2) |
| 10 | 0.1003 | ยฝ ln(11/9) |
Frequently Asked Questions
arcoth(x) = artanh(1/x) for |x| > 1. They share the same derivative formula 1/(1โxยฒ), but are defined on complementary domains.
Because hyperbolic cotangent coth(t) only produces values with |coth(t)| > 1. As t approaches 0, coth(t) approaches ยฑโ, and as t approaches ยฑโ, coth(t) approaches ยฑ1 but never reaches it.
Use Math.atanh(1/x) in JavaScript or math.atanh(1/x) in Python. Or use the formula: 0.5 * Math.log((x+1)/(x-1)).
arcoth(2) โ 0.5493. This equals ยฝ ln(3), since (2+1)/(2โ1) = 3.
Inverse hyperbolic cotangent appears in fluid dynamics, heat transfer equations, and certain integral solutions involving partial fractions.